I have computed the infinite multiplication using Maple, $\Pi_{k=3}^{\infty} ( \cos (\frac{\pi}{k} ))$, as follows, but it resulted in **0**! I wonder why this happened, although maple was using exact arithmetic.

P := Product(cos(Pi/k), k = 3 .. infinity)

value(P)

Note that if I use floating-point instead, it gives me the right answer, **0.1149420449**.

evalf(P)

By the way, I did not expect such a situation! Exact arithmetic should be exact! I am multiplying non-zero numbers to each other, starting from $1/2$ to $1$ as $n \rightarrow \infty$. So it should not be zero! *Why such a thing happened?*